L5 Continuity - Practice 1
Completion requirements
Unit 1B
Limits
Lesson 5: Continuity
Practice
Once you feel confident with continuous and discontinuous functions, click on the Practice tab and complete problems 1 to 3. Check your answers by going to the Solutions tab.
Instructions: Click the Download File button to download a printable PDF of the questions. Answer each of the following practice questions on a separate piece of paper. Step by step solutions are provided under the Solutions tab. You will learn the material more thoroughly if you complete the questions before checking the answers.
Instructions: Click the Download File button to download a printable PDF of the questions. Answer each of the following practice questions on a separate piece of paper. Step by step solutions are provided under the Solutions tab. You will learn the material more thoroughly if you complete the questions before checking the answers.
1.
The graph of the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» is shown below. Determine
where the function is discontinuous.
2.
The graph of the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»h«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» is shown below. Evaluate
each of the following limits, if they exist.
a.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«msup»«mn»1«/mn»«mo»§#8722;«/mo»«/msup»«/mrow»«/munder»«mi»h«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»
b.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«msup»«mn»1«/mn»«mo»+«/mo»«/msup»«/mrow»«/munder»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»
c.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«mn»1«/mn»«/mrow»«/munder»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»
d.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»
e.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»3«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»
f.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»
3.
Where is the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»4«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» discontinuous?
1.
The graph of the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» is shown below.
Determine where the function is discontinuous.
The function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» is discontinuous on
the interval «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»-«/mo»«mn»2«/mn»«mo»§#60;«/mo»«mi»x«/mi»«mo»§#60;«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math» because
there is a break in the graph.
2.
The graph of the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»h«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» is shown below.
Evaluate each of the following limits, if they exist.
a.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«msup»«mn»1«/mn»«mo»§#8722;«/mo»«/msup»«/mrow»«/munder»«mi»h«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»
b.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«msup»«mn»1«/mn»«mo»+«/mo»«/msup»«/mrow»«/munder»«mi»h«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»
c.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«mn»1«/mn»«/mrow»«/munder»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»
d.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»
e.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»3«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»
f.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»
a.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«msup»«mn»1«/mn»«mo»§#8722;«/mo»«/msup»«/mrow»«/munder»«mi»h«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math»
b.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«msup»«mn»1«/mn»«mo»+«/mo»«/msup»«/mrow»«/munder»«mi»h«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math»
c.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«mn»1«/mn»«/mrow»«/munder»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi»DNE«/mi»«/mstyle»«/math»
d.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math»
e.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»3«/mn»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math»
f.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math»
3.
Where is the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»4«/mn»«msup»«mi
mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» discontinuous?
The non-permissible value is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math». This non-permissible value will either cause a point of discontinuity
or a break in the graph. The function is discontinuous at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math»
as shown in the graph below.
